Tunneling through a parabolic barrier viewed from Wigner phase space

TL;DR

This paper investigates quantum tunneling through a parabolic barrier using the Wigner phase space formalism, providing a phase space perspective on reflection and transmission coefficients for an inverted harmonic oscillator.

Contribution

It introduces a phase space approach to analyze tunneling, explicitly solving for the Wigner function of energy eigenstates in an inverted oscillator potential.

Findings

Derived expressions for reflection and transmission coefficients in phase space

Provided a detailed phase space analysis of tunneling phenomena

Connected classical trajectories with quantum tunneling probabilities

Abstract

We analyze the tunneling of a particle through a repulsive potential resulting from an inverted harmonic oscillator in the quantum mechanical phase space described by the Wigner function. In particular, we solve the partial differential equations in phase space determining the Wigner function of an energy eigenstate of the inverted oscillator. The reflection or transmission coefficients $R$ or $T$ are then given by the total weight of all classical phase space trajectories corresponding to energies below, or above the top of the barrier given by the Wigner function.